%I #15 Jul 16 2015 22:22:15

%S 2,1,5,4,19,17,16,75,67,66,64,269,263,266,257,256,1053,1031,1035,1029,

%T 1026,1024,4125,4119,4123,4107,4099,4098,4096,16479,16427,16431,16407,

%U 16395,16391,16386,16384,65709,65629,65579,65581,65559,65543,65539,65537,65536

%N Minimal number (in decimal representation) with n nonprime substrings in base-4 representation (substrings with leading zeros are considered to be nonprime).

%C The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty. Proof: Define m(n):=2*sum_{j=i..k} 4^j, where k:=floor((sqrt(8*n+1)-1)/2), i:= n-A000217(k). For n=0,1,2,3,... the m(n) in base-4 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)*(k+2)/2)-k-1+i = (k*(k+1)/2)+i = n, which proves the statement.

%C If p is a number with k prime substrings and d digits (in base-4 representation), m>=d, than b := p*4^(m-d) has m*(m+1)/2 - k nonprime substrings, and a(A000217(n)-k) <= b.

%H Hieronymus Fischer, <a href="/A217104/b217104.txt">Table of n, a(n) for n = 0..528</a>

%F a(n) >= 4^floor((sqrt(8*n-7)-1)/2) for n>0, equality holds if n is a triangular number (cf. A000217).

%F a(A000217(n)) = 4^(n-1), n>0.

%F a(A000217(n)-k) >= 4^(n-1) + k, 0<=k<n, n>0.

%F a(A000217(n)-k) = 4^(n-1) + p, where p is the minimal number >= 0 such that 4^(n-1) + p, has k prime substrings in base-4 representation, 0<=k<n, n>0.

%e a(0) = 2, since 2 = 2_4 is the least number with zero nonprime substrings in base-4 representation.

%e a(1) = 1, since 1 = 1_4 is the least number with 1 nonprime substring in base-4 representation.

%e a(2) = 5, since 5 = 11_4 is the least number with 2 nonprime substrings in base-4 representation (these are 2-times 1).

%e a(3) = 4, since 4 = 10_4 is the least number with 3 nonprime substrings in base-4 representation (these are 0, 1 and 10).

%e a(4) = 19, since 19 = 103_4 is the least number with 4 nonprime substrings in base-4 representation, these are 0, 1, 10, and 03 (remember, that substrings with leading zeros are considered to be nonprime).

%e a(7) = 75, since 75 = 1023_4 is the least number with 7 nonprime substrings in base-4 representation, these are 0, 1, 10, 02, 023, 102 and 1023 (remember, that substrings with leading zeros are considered to be nonprime: 2_4 = 2, 3_4 = 3 and 23_4 = 11 are the only base-4 prime substrings of 75).

%Y Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.

%Y Cf. A035244, A079397, A213300-A213321.

%Y Cf. A217102-A217109.

%Y Cf. A217302-A217309.

%K nonn,base

%O 0,1

%A _Hieronymus Fischer_, Dec 12 2012